We study properties of the
truncated kernel function ![$ \gamma_2$](img2.gif)
defined on integers
![$ n\ge 2$](img3.gif)
by
![$ \gamma_2(n)=\gamma(n)/P(n)$](img4.gif)
, where
![$ \gamma(n)=\prod_{p\vert n}p$](img5.gif)
is the well-known
kernel function and
![$ P(n)$](img6.gif)
is the largest prime factor of
![$ n$](img7.gif)
. In particular, we show that the maximal order of
![$ \gamma_2(n)$](img8.gif)
for
![$ n\le x$](img9.gif)
is
![$ (1+o(1))x/\log x$](img10.gif)
as
![$ x\to \infty$](img11.gif)
and that
![$ \sum_{n\le x} 1/\gamma_2(n)= (1+o(1)) \eta x/\log x$](img12.gif)
, where
![$ \eta=\zeta(2)\zeta(3)/\zeta(6)$](img13.gif)
. We further show that, given any positive real number
![$ u<1$](img14.gif)
,
![$ \lim_{x\to \infty} \frac 1x \char93 \{n\le x: \gamma_2(n)<x^u\}=
\lim_{x\to \infty} \frac 1x \char93 \{n\le x: n/P(n) < x^u\}
= 1-\rho(1/(1-u))$](img15.gif)
, where
![$ \rho$](img16.gif)
is the Dickman function.
We also show that
![$ n/P(n)$](img17.gif)
can very often be much larger than
![$ \gamma_2(n)$](img8.gif)
, namely by proving that, given any
![$ c\in [1,\xi)$](img18.gif)
, where
![$ \xi$](img19.gif)
is the unique solution to
![$ \xi\log 2 = \log(1+\xi)+\xi \log(1+1/\xi)$](img20.gif)
, then